viscosity.F90

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!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!
!  Function :  v i s c o s i t y
!
!> @file
!!
!! Ice viscosity as a function of the effective strain rate and the temperature
!! (in cold ice) or the water content (in temperate ice) or both (for the
!! enthalpy method).
!!
!! @section Copyright
!!
!! Copyright 2009-2016 Ralf Greve
!!
!! @section License
!!
!! This file is part of SICOPOLIS.
!!
!! SICOPOLIS is free software: you can redistribute it and/or modify
!! it under the terms of the GNU General Public License as published by
!! the Free Software Foundation, either version 3 of the License, or
!! (at your option) any later version.
!!
!! SICOPOLIS is distributed in the hope that it will be useful,
!! but WITHOUT ANY WARRANTY; without even the implied warranty of
!! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
!! GNU General Public License for more details.
!!
!! You should have received a copy of the GNU General Public License
!! along with SICOPOLIS.  If not, see <http://www.gnu.org/licenses/>.
!<
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

!-------------------------------------------------------------------------------
!> Ice viscosity as a function of the effective strain rate and the temperature
!! (in cold ice) or the water content (in temperate ice) or both (for the
!! enthalpy method).
!<------------------------------------------------------------------------------
function viscosity(de_val, temp_val, temp_m_val, omega_val, enh_val, &
                   i_flag_cold_temp)

use sico_types
use sico_variables
use sico_vars
use ice_material_quantities, only : ratefac_c, ratefac_t, ratefac_c_t

implicit none

real(dp)                 :: viscosity
real(dp)    , intent(in) :: de_val
real(dp)    , intent(in) :: temp_val, temp_m_val
real(dp)    , intent(in) :: omega_val
real(dp)    , intent(in) :: enh_val
integer(i2b), intent(in) :: i_flag_cold_temp

real(dp) :: ratefac_val
real(dp) :: de_val_m

real(dp) :: n_power_law, inv_n_power_law, n_grain_size

real(dp), parameter :: de_min = 1.0e-30_dp   ! minimum value for the
                                             ! effective strain rate

#if (FLOW_LAW==4)
real(dp), parameter :: sm_coeff_1 = 8.5112e-15_dp, &   ! s^-1 Pa^-1
                       sm_coeff_2 = 8.1643e-25_dp, &   ! s^-1 Pa^-3
                       sm_coeff_3 = 9.2594e-12_dp      ! Pa^-2
#endif

!-------- Rate factor and effective strain rate --------

if (i_flag_cold_temp == 0_i2b) then   ! cold ice
   ratefac_val = ratefac_c(temp_val, temp_m_val)
else if (i_flag_cold_temp == 1_i2b) then   ! temperate ice
   ratefac_val = ratefac_t(omega_val)
else   ! enthalpy method
   ratefac_val = ratefac_c_t(temp_val, omega_val, temp_m_val)
end if

de_val_m = max(de_val, de_min)

#if (FIN_VISC==1)

#if (FLOW_LAW==1)

!-------- Glen's flow law (n=3) --------

inv_n_power_law = 0.333333333333333_dp   ! 1/3

viscosity = 0.5_dp * de_val_m**(inv_n_power_law-1.0_dp) &
                   * (enh_val*ratefac_val)**(-inv_n_power_law)

#elif (FLOW_LAW==2)

!-------- Goldsby-Kohlstedt flow law (n=1.8, d=1.4) --------

inv_n_power_law = 0.555555555555555_dp   ! 1/1.8
n_grain_size    = 1.4_dp

viscosity = 0.5_dp * de_val_m**(inv_n_power_law-1.0_dp) &
                   * gr_size**(n_grain_size*inv_n_power_law) &
                   * (enh_val*ratefac_val)**(-inv_n_power_law)

#elif (FLOW_LAW==3)

!-------- Durham's flow law (n=4) --------

inv_n_power_law = 0.25_dp   ! 1/4

viscosity = 0.5_dp * de_val_m**(inv_n_power_law-1.0_dp) &
                   * (enh_val*ratefac_val)**(-inv_n_power_law)

#endif

#elif (FIN_VISC==2)

#if (FLOW_LAW==1)

!-------- Glen's flow (n=3) with additional finite viscosity --------

n_power_law = 3.0_dp

viscosity = visc_iter(de_val_m, ratefac_val, enh_val, n_power_law, sigma_res)

#elif (FLOW_LAW==2)

!-------- Goldsby-Kohlstedt flow law (n=1.8, d=1.4)
!                           with additional finite viscosity --------

n_power_law  = 1.8_dp
n_grain_size = 1.4_dp

write(6,'(a)') ' viscosity: Computation of the viscosity as a function'
write(6,'(a)') '            of the effective strain rate not yet implemented'
write(6,'(a)') '        for grain-size-dependent finite viscosity flow laws!'
stop 

#elif (FLOW_LAW==3)

!-------- Durham's flow law (n=4) with additional finite viscosity --------

n_power_law = 4.0_dp

viscosity = visc_iter(de_val_m, ratefac_val, enh_val, n_power_law, sigma_res)

#endif

#endif

#if (FLOW_LAW==4)

!-------- Smith-Morland (polynomial) flow law --------

viscosity = visc_iter_sm(de_val_m, ratefac_val, enh_val, &
                         sm_coeff_1, sm_coeff_2, sm_coeff_3)

#endif

end function viscosity

#if (FIN_VISC==2)

!-------------------------------------------------------------------------------
!> Iterative computation of the viscosity by solving equation (4.28)
!! by Greve and Blatter (Springer, 2009).
!<------------------------------------------------------------------------------
function visc_iter(de_val_m, ratefac_val, enh_val, n_power_law, sigma_res)

use sico_types

implicit none

real(dp)             :: visc_iter
real(dp), intent(in) :: de_val_m
real(dp), intent(in) :: ratefac_val, enh_val
real(dp), intent(in) :: n_power_law, sigma_res

real(dp) :: visc_val, res

real(dp), parameter :: eps = 1.0e-05_dp   ! convergence parameter

!-------- Determination of the order of magnitude --------

visc_val = 1.0e+10_dp   ! initial guess (very low value)

do

   res = fct_visc(de_val_m, ratefac_val, enh_val, visc_val, &
                  n_power_law, sigma_res)

   if (res < 0.0_dp) then
      visc_val = 10.0_dp*visc_val
   else
      exit
   end if

end do

!-------- Newton's method --------

do

   visc_val = visc_val &
              - res &
                /fct_visc_deriv(de_val_m, ratefac_val, enh_val, visc_val, &
                                n_power_law, sigma_res)

   res = fct_visc(de_val_m, ratefac_val, enh_val, visc_val, &
                  n_power_law, sigma_res)
                  
   if (abs(res) < eps) exit

end do

visc_iter = visc_val

end function visc_iter

!-------------------------------------------------------------------------------
!> Viscosity polynomial
!! [equation (4.28) by Greve and Blatter (Springer, 2009)].
!<------------------------------------------------------------------------------
function fct_visc(de_val_m, ratefac_val, enh_val, visc_val, &
                  n_power_law, sigma_res)

use sico_types

implicit none

real(dp)             :: fct_visc
real(dp), intent(in) :: de_val_m
real(dp), intent(in) :: ratefac_val, enh_val
real(dp), intent(in) :: visc_val
real(dp), intent(in) :: n_power_law, sigma_res

fct_visc = 2.0_dp**n_power_law &
             *enh_val*ratefac_val &
             *de_val_m**(n_power_law-1.0_dp) &
             *visc_val**n_power_law &
          + 2.0_dp*enh_val*ratefac_val &
             *sigma_res**(n_power_law-1.0_dp) &
             *visc_val &
          - 1.0_dp

end function fct_visc

!-------------------------------------------------------------------------------
!> Derivative of the viscosity polynomial
!! [equation (4.28) by Greve and Blatter (Springer, 2009)].
!<------------------------------------------------------------------------------
function fct_visc_deriv(de_val_m, ratefac_val, enh_val, visc_val, &
                        n_power_law, sigma_res)

use sico_types

implicit none

real(dp)             :: fct_visc_deriv
real(dp), intent(in) :: de_val_m
real(dp), intent(in) :: ratefac_val, enh_val
real(dp), intent(in) :: visc_val
real(dp), intent(in) :: n_power_law, sigma_res

fct_visc_deriv = 2.0_dp**n_power_law*n_power_law &
                   *enh_val*ratefac_val &
                   *de_val_m**(n_power_law-1.0_dp) &
                   *visc_val**(n_power_law-1.0_dp) &
                 + 2.0_dp*enh_val*ratefac_val &
                   *sigma_res**(n_power_law-1.0_dp)
         
end function fct_visc_deriv

#endif

#if (FLOW_LAW==4)

!-------------------------------------------------------------------------------
!> Iterative computation of the viscosity by solving equation (4.33)
!! [analogous to (4.28)] by Greve and Blatter (Springer, 2009).
!<------------------------------------------------------------------------------
function visc_iter_sm(de_val_m, ratefac_val, enh_val, &
                      sm_coeff_1, sm_coeff_2, sm_coeff_3)

use sico_types

implicit none

real(dp)             :: visc_iter_sm
real(dp), intent(in) :: de_val_m
real(dp), intent(in) :: ratefac_val, enh_val
real(dp), intent(in) :: sm_coeff_1, sm_coeff_2, sm_coeff_3

real(dp) :: visc_val, res

real(dp), parameter :: eps = 1.0e-05_dp   ! convergence parameter

!-------- Determination of the order of magnitude --------

visc_val = 1.0e+10_dp   ! initial guess (very low value)

do

   res = fct_visc_sm(de_val_m, ratefac_val, enh_val, visc_val, &
                     sm_coeff_1, sm_coeff_2, sm_coeff_3)

   if (res < 0.0_dp) then
      visc_val = 10.0_dp*visc_val
   else
      exit
   end if

end do

!-------- Newton's method --------

do

   visc_val = visc_val &
              - res &
                /fct_visc_sm_deriv(de_val_m, ratefac_val, enh_val, visc_val, &
                                   sm_coeff_1, sm_coeff_2, sm_coeff_3)

   res = fct_visc_sm(de_val_m, ratefac_val, enh_val, visc_val, &
                     sm_coeff_1, sm_coeff_2, sm_coeff_3)
                  
   if (abs(res) < eps) exit

end do

visc_iter_sm = visc_val
         
end function visc_iter_sm

!-------------------------------------------------------------------------------
!> Viscosity polynomial
!! [equation (4.33) by Greve and Blatter (Springer, 2009)].
!<------------------------------------------------------------------------------
function fct_visc_sm(de_val_m, ratefac_val, enh_val, visc_val, &
                     sm_coeff_1, sm_coeff_2, sm_coeff_3)

use sico_types

implicit none

real(dp)             :: fct_visc_sm
real(dp), intent(in) :: de_val_m
real(dp), intent(in) :: ratefac_val, enh_val
real(dp), intent(in) :: visc_val
real(dp), intent(in) :: sm_coeff_1, sm_coeff_2, sm_coeff_3

real(dp) :: de_visc_factor

de_visc_factor = de_val_m*de_val_m*visc_val*visc_val
                   
fct_visc_sm = 2.0_dp*enh_val*ratefac_val*visc_val &
              * ( sm_coeff_1 &
                  + 4.0_dp*sm_coeff_2*de_visc_factor &
                    * ( 1.0_dp + 4.0_dp*sm_coeff_3*de_visc_factor ) ) &
              - 1.0_dp

end function fct_visc_sm

!-------------------------------------------------------------------------------
!> Derivative of the viscosity polynomial
!! [equation (4.33) by Greve and Blatter (Springer, 2009)].
!<------------------------------------------------------------------------------
function fct_visc_sm_deriv(de_val_m, ratefac_val, enh_val, visc_val, &
                           sm_coeff_1, sm_coeff_2, sm_coeff_3)

use sico_types

implicit none

real(dp)             :: fct_visc_sm_deriv
real(dp), intent(in) :: de_val_m
real(dp), intent(in) :: ratefac_val, enh_val
real(dp), intent(in) :: visc_val
real(dp), intent(in) :: sm_coeff_1, sm_coeff_2, sm_coeff_3

real(dp) :: de_visc_factor

real(dp), parameter :: twenty_over_three = 6.666666666666667_dp
                          
de_visc_factor = de_val_m*de_val_m*visc_val*visc_val
                          
fct_visc_sm_deriv = 2.0_dp*sm_coeff_1*enh_val*ratefac_val &
                   + 24.0_dp*sm_coeff_2*enh_val*ratefac_val*de_visc_factor &
                     * ( 1.0_dp + twenty_over_three*sm_coeff_3*de_visc_factor )
         
end function fct_visc_sm_deriv

#endif
!